A concept of discrete space-time was first suggested by V. Ambartsumian and D. Ivanenko in 1930. This concept has not been formulated in a strict mathematical form, but often is related to a hypothesis of a minimal fundamental length, e. g., in non-linear field theories with self-interaction constants of dimension of length. One usually considers a lattice in coordinate space and introduces a cut-off factor in a momentum space.

An acceptable mathematical formalization of a discrete space-time can be given in terms of topological spaces

*Y*where a connected component of any point*y*of*Y*is the closure of*y*. For instance, if*Y*is a Hausdorff space, then a connected component of its point*y*is this point itself, i. e.,*Y*is a totally disconnected space.This is the case of a discrete topological space, the space of rational numbers, vector spaces and analytic manifolds over fields with ultra-metric absolute values. In algebraic quantum field theory, the spectrum of some C*-algebras, e. g., the C*-algebras of probability measures, is totally disconnected.

**References:**

G. Sardanashvily, Discrete space-time,

*Enciclopaedia of Mathematics*(Springer)D.Ivanenko, G.Sardanashvily, Towards a model of discrete space-time,

*Russ. Phys. J*.**21**(1978) 1508.
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